1. The problem statement, all variables and given/known data For the differential equation, verify (by differentiation and substitution) that the given function y(t) is a solution. 2. Relevant equations [itex] y' - 4ty = 1 [/itex] [itex] y(t) = \int_{0}^{t} e^{-2(s^{2}-t^{2})} ds[/itex] 3. The attempt at a solution I attempted to take [itex]\frac{d}{dt}[/itex] of y(t) as usual but 1. if I do not try bringing the [itex]\frac{d}{dt}[/itex] inside the integral I can do nothing because there is no elementary antiderivative of y(t). 2. if I do bring the [itex]\frac{d}{dt}[/itex] inside the integral, I can use the chain rule to get [itex] y(t) = (-2) \int_{0}^{t} (-2t) e^{-2(s^{2}-t^{2})} ds[/itex] but since my variable of integration is ds not dt, this doesn't allow me to use a u-substitution as I had hoped nor can I think of a way to relate ds and dt. More or less I do not know how to take [itex]\frac{d}{dt}[/itex] of y(t) and I do not know any other ways to solve the problem.
I don't think you have to be able to do the integral to identify that expression as [itex]4 t y(t)[/itex]. Since you are integrating ds you can factor out the t.